Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #5464 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3×C42 |
| direct product | C2×C3×#5464 |
| Fitting | C2×C3×H(3,𝔽4) ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C2×C32 |
| Sylow | C2×H(3,𝔽4) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #5464 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C32 |
| Frattini | C22 | C2×C3×C22⋊A4 |
| Fitting | C2×C3×H(3,𝔽4) | C3 |
| socle | C23×C3 | C22⋊A4 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C2×C3 | |
640 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 640 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5464, C2×C3×#3532⋊C3, C2×C3×#85559, … |
| maximal normal subgroup of | C2×C3×#3532⋊C3, C2×C3×#85559, C2×C3×#5450⋊C3 |
| maximal quotient of | AGL(1,𝔽7)×#5464, C2×C3×#2679⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153847);